Notes on questions about spaces with algebraic structures

In this paper, we discuss properties of topological spaces with algebraic structures and answer several problems posed in [A.V. Arhangelʼskii, M. Tkachenko, Topological Groups and Related Structures, Atlantics Press/World Sci., 2008]. We consider a topology on an infinite discrete group G generated by a free ultrafilter p on G and show that this topology can be Hausdorff in the case when G is the group of integers, even if p is not an idempotent. Two open continuous homomorphisms f of a paratopological group G onto a paratopological group H are constructed such that: (a) H is paracompact and the kernel of f is locally compact, but f is not locally perfect and G is not locally paracompact; (b) H and the kernel of f are metrizable, but G is not metrizable. We also show that every first-countable ω-narrow semitopological group is separable.